Weighted Sum Formulas Research Articles

ON GENERALIZATIONS OF WEIGHTED SUM FORMULAS OF MULTIPLE ZETA VALUES

In this paper, we compute shuffle relations from multiple zeta values of the form ζ({1}m-1, n+1) or sums of multiple zeta values of fixed weight and depth. Some interesting weighted sum formulas are obtained, such as [Formula: see text] where m and k are positive integers with m ≥ 2k. For k = 1, this gives Ohno–Zudilin's weighted sum formula.

WEIGHTED SUMS WITH TWO PARAMETERS OF MULTIPLE ZETA VALUES AND THEIR FORMULAS

A typical formula of multiple zeta values is the sum formula which expresses a Riemann zeta value as a sum of all multiple zeta values of fixed weight and depth. Recently weighted sum formulas, which are weighted analogues of the sum formula, have been studied by many people. In this paper, we give two formulas of weighted sums with two parameters of multiple zeta values. As applications of the formulas, we find some linear combinations of multiple zeta values which can be expressed as polynomials of usual zeta values with coefficients in the rational polynomial ring generated by the two parameters, and obtain some identities for weighted sums of multiple zeta values.

Model and methods for comprehensive measurement of the low-carbon status of China’s oil and gas enterprises

This paper establishes a model that would allow China’s oil and gas enterprises to scientifically evaluate and measure their low-carbon level and status. It considers various characteristics of China’s oil and gas enterprises and the implications of low-carbon development, and is based on an overall analysis of factors that influence the reduction of carbon emissions. In view of low-carbon economic theories and the general principles of an evaluation index system, a comprehensive system for measuring the low-carbon status of China’s oil and gas enterprises has been developed. This measurement system is comprised of four main criteria (energy structure, energy utilization, carbon emissions and utilization, and low carbon management) as well as thirty indexes. By the Delphi method and the analytical hierarchy process (AHP), the weight of the rules hierarchy and indexes hierarchy were determined. The standardized indexes were then integrated using a linear weighted sum formula, and a comprehensive formula for index measurement was established. Taking into account the status of low-carbon development in the petroleum and petrochemical industry at home and abroad, an evaluation criterion is proposed comprising four levels: ideal low-carbon, economical low-carbon, medium-carbon and high-carbon, whose values were organized within the settings of [0,1].

Weighted sum formula for multiple zeta values

The sum formula is a basic identity of multiple zeta values that expresses a Riemann zeta value as a homogeneous sum of multiple zeta values of a given dimension. This formula was already known to Euler in the dimension two case, conjectured in the early 1990s for higher dimensions and then proved by Granville and Zagier. Recently a weighted form of Euler's formula was obtained by Ohno and Zudilin. We generalize it to a weighted sum formula for multiple zeta values of all dimensions.

Reply to Comments on the Nature and Uses of Expectancy-Value Models in Consumer Attitude Research

1. In their enthusiasm to relate other theories to Fishbein's, the authors have misunderstood and misinterpreted Rosenberg and the antecedent functional (means-end) school of thought based on the concept of valued states. 2. Contrary to assertions made by Fishbein, I don't think the Fishbein and Rosenberg models are equivalent or even similar in structure. The illusion of an equivalent structure comes from the weightedsum formula used by both to obtain an index of cognitive structure. 3. There is no Bass-Talarzyk-Sheth theory of attitude. However, a comprehensive theory of attitude-behavior relationship developed by Sheth some time ago is known to all three authors. 4. It is indeed naive to assert superiority of one model over other models when the studies (1) reflected different settings (e.g., laboratory experiment vs. field survey), (2) involved different issues (e.g., Negroes should be allowed in white neighborhood, story about Mrs. Williams, brands of grocery products), and (3) utilized different statistical and methodological procedures (e.g., chi-squared, simple correlations, multiple correlations).

Application of weighted phase-sum formulae for phase determination

The simple `weighted sum formula' has been studied as regards its applicability to phase refinements. These studies show that proper weighting of the phase relations significantly improves the phase estimates from phase-sum formulae. The evaluation of weights used in the `variance-weighted sum formula' is discussed, together with the problem of the 2π ambiguity. A simple procedure for selecting efficient basis sets is also given. The procedures described have been programmed and used to solve twelve unknown structures. Applications to somewhat complicated cases are described.